Density function question

Question

Suppose $x_{1}, x_{2} \dots x_{N}$ are gaussian RVs with variance $S$ and mean $1$. What is the density function of

$$\frac{ |\sum_{n=1}^{N}x_{n}|^{2}}{\sum_{n=1}^{N}|x_{n}|^{2}}\text{?}$$

Answer 1

For the case $N=3,\;x_i\sim\mathcal{N}(\mu,\sigma),\;i\in\{1,2,3\}$, i calculated a CDF of

$P(y\leq\alpha)=\text{exp}\Big(-\frac{1}{2}(3-\alpha)\big(\frac{\mu}{\sigma}\big)^2\Big)\sqrt{\alpha/3}\;,\quad y=\frac{(x_1+x_2+x_3)^2}{x_1^2+x_2^2+x_3^2},\quad 0\leq\alpha\leq N$.

The other cases $N\neq 3$ are much harder to solve (at least for me). An idea that helped the solution was rotating the coordinate system so that the former $(x_1, x_2, x_3)\propto(1,1,1)$-direction is aligned to an axis from the new coordinate system, e.g. $(z_1, z_2, z_3)\propto(0,0,1)$. Then

$y=\frac{(x_1+x_2+x_3)^2}{x_1^2+x_2^2+x_3^2}=\frac{3 z_3^2}{z_1^2+z_2^2+z_3^2}$,

which makes integration easier.